# complex limits

Given that $z$ is a complex number of the form $z = a +b i$, where $a, \, b \in \mathbb{R}$, and $\overline{z}$ is its conjugate, what is $$\lim_{z \to 0} \frac{\left( \overline{z} \right)^2}{ z^2} =\text{?}$$

When I ask Sage to do this limit, it says it's 1. I am almost certain that's not right. Here's the Sage code:

sage: z=var('z')

sage: assume(z,'complex')

sage: limit((conjugate(z))^2/z^2,z=0)

My analysis leads to DNE, so I wonder if I am misusing Sage.

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The limit is equivalent to two limits of functions of two real variables

x,y=var('x y',domain=RR)
z=x+I*y
f=(z.conjugate()/z)^2
f,  f.rectform()
((x - I*y)^2/(x + I*y)^2,
-4*I*(x^2 - y^2)*x*y/(x^2 + y^2)^2 - (4*x^2*y^2 - (x^2 - y^2)^2)/(x^2 + y^2)^2)


It does not exist, since the limits on some subsets are different

limit(f.subs(y=0),x=0),  limit(f.subs(x=y),y=0)
(1, -1)


Use

plot3d(real_part(f),(x,-1,1),(y,-1,1))
plot3d(imag_part(f),(x,-1,1),(y,-1,1))


for better understanding

Using another approach, if a is the modulus and b the argument, then

a,b=var('a b',domain=RR)
z=a*exp(I*b)
f=(z.conjugate()/z)^2;f

e^(-4*I*b)


and the limit

limit(f,a=0)
e^(-4*I*b)


depends on the argument , so the limit with z->0 does not exist

more

( 2022-12-03 11:33:13 +0200 )edit

Another way to see it is to search the directional limit of f :

sage: var("rho, theta", domain="real")
(rho, theta)
sage: f=rho*(cos(theta)+I*sin(theta))
sage: (f.conjugate()^2/f^2).limit(rho=0)._sympy_().simplify().rewrite("sin")._sage_()
cos(4*theta) - I*sin(4*theta)


which show that this limit depends of the direction of your trajectory to 0...

BTW :

sage: (f.conjugate()^2/f^2)._sympy_().simplify().rewrite("sin")._sage_()
cos(4*theta) - I*sin(4*theta)


which does not depend on rho...

( 2022-12-03 13:26:59 +0200 )edit

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Last updated: Dec 03 '22